Detecting free summands via vanishing of (co)homology, and applications to Berger's conjecture

Abstract

This work presents some of the results from the preprint [3]. It deals with the vanishing of (co)homology and some of its implications in module theory over commutative Noetherian local rings, specifically in the detection of free direct summands and the establishment of freeness criteria. Both types of results are desirable and are closely connected to the recent literature, not only for their intrinsic relevance but also for providing fundamental insights, such as the characterization of Gorenstein and regularity properties for local rings. Regarding the latter, the work contributes to the celebrated Berger s Conjecture, which connects, in dimension 1, the torsion of the module of Kähler differentials to its freeness, and consequently, to the regularity of the given local ring.